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arXiv:2005.11521 · Communications in Theoretical Physics

Total variance and invariant information in complementary measurements

Bin Chen, Shao-Ming Fei

Published May 22, 2020• 10 citations
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Abstract

Abstract We investigate the total variance of a quantum state with respect to a complete set of mutually complementary measurements and its relation to the Brukner–Zeilinger invariant information. By summing the variances over any complete set of mutually unbiased measurements and general symmetric informationally complete measurements respectively, we show that the Brukner–Zeilinger invariant information associated with such types of quantum measurements is equal to the difference between the maximal variance and the total variance obtained. These results provide an operational link between the previous interpretations of the Brukner–Zeilinger invariant information.

DOI
10.1088/1572-9494/ab8a18

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Abstract We investigate the total variance of a quantum state with respect to a complete set of mutually complementary measurements and its relation to the Brukner–Zeilinger invariant information. By summing the variances over any complete set of mutually unbiased measurements and general symmetric informationally complete measurements respectively, we show that the Brukner–Zeilinger invariant information associated with such types of quantum measurements is equal to the difference between the maximal variance and the total variance obtained. These results provide an operational link between the previous interpretations of the Brukner–Zeilinger invariant information.
We address the problem of properly quantifying information in quantum theory. Brukner and Zeilinger proposed the concept of an operationally invariant measure based on measurement statistics. Their measure of information is calculated with probabilities generated in a complete set of mutually complementary observations. This approach was later criticized for several reasons. We show that some critical points can be overcome by means of natural extension or reformulation of the Brukner–Zeilinger approach. In particular, this approach is connected with symmetric informationally complete measurements. The ‘total information’ of Brukner and Zeilinger can further be treated in the context of mutually unbiased measurements as well as general symmetric informationally complete measurements. The Brukner–Zeilinger measure of information is also examined in the case of detection inefficiencies. It is shown to be decreasing under the action of bistochastic maps. The Brukner–Zeilinger total information can be used for estimating the map norm of quantum operations.
Abstract Brukner–Zeilinger invariants, including invariant information and invariant uncertainty, play a remarkable role in quantum information theory. Recently, a broad class of symmetric informationally complete measurements is presented, which contains a general symmetrical information complete POVMs and mutually unbiased measurements (2022, Phys. Rev. A 105, 042 209). Based on this important measurement and variance, we further analyze the internal relationship between Brukner–Zeilinger invariants in detail. The results maybe provide a method to explain the importance of Brukner–Zeilinger invariants.
To quantify the intrinsic information content in a quantum state, Brukner and Zeilinger introduced the concept of operationally invariant information in terms of the outcome probabilities of measuring a complete set of mutually complementary observables [\ifmmode \check{C}\else \v{C}\fi{}. Brukner and A. Zeilinger, Phys. Rev. Lett. 83, 3354 (1999)]. This information quantity has basic significance and implications, and the present work is devoted to some further studies of it. We first introduce the Brukner-Zeilinger invariant information in the presence of conjugate symmetry or antisymmetry, which are motivated by considerations of fundamental issues concerning conjugate symmetry in quantum mechanics. Then we prove that both the Brukner-Zeilinger invariant information with conjugate symmetry and that with conjugate antisymmetry are convex in the quantum state, and we show that they constitute a natural decomposition of the Brukner-Zeilinger invariant information. We further relate them to the imaginarity (i.e., the usage of a complex number field) of quantum mechanics and evaluate their extreme values.
Abstract In order to measure the inherent information content in a quantum state, Brukner and Zeilinger developed the idea of operationally invariant information in terms of the outcome probabilities of measuring a whole collection of mutually complementary observables. In this work, by partitioning the orthonormal operator basis of the associated operator Hilbert space into two complementary subsets, we establish a new decomposition of the Brukner-Zeilinger information. We further demonstrate that the two complementary components of the Brukner-Zeilinger information capture nicely the wave and particle features encoded in the quantum state after an n -path interferometer. Thus, we establish a universal complementarity relation involving the interference (wave feature) and the predictability (particle feature) via the unified concept of the Brukner-Zeilinger invariant information. These findings provide new characterizations of the wave-particle duality via the fundamental Brukner-Zeilinger invariant information.
Observables in quantum mechanics are generally complementary, that is, they reveal mutually incompatible pieces of information about a given system. This property is not only a fundamental tenet of the quantum formalism, but also a key component in quantum cryptographic protocols. As such it has fuelled much research into finding sets of highly complementary observables. In its strongest form -- the one we consider in this work -- the information between complementary observables is not merely incompatible but mutually exclusive: maximal information about one observable implies no information about the other, and vice versa. Maximal sets of non-degenerate complementary observables are known to exist in systems of prime power dimension such as n-qubit systems. Here, we study complementarity of degenerate observables, specifically we prove that observables associated with the n-qubit Pauli group also exhibit complementarity under this restriction: first, we obtain a criterion for two such observables to be complementary and, second, we relate maximal sets of complementary observables with informational pure state complementarity equalities, further studied in two companion papers. Equivalently, these results can be formulated in terms of maximal complementary sets of (not necessarily maximal) Abelian subgroups of the Pauli group linking with (possibly coarse-grained) mutually unbiased bases. Finally, we prove that these complementarity sets entail strong entropic uncertainty relations.